Wikipedia claims that for a given group $G$ with the identity element $e$ the commutator $[e,e]=e$ is the only commutator if and only if $G$ is Abelian.
I know that for a given $N \trianglelefteq G$ the quotient $G/N$ is Abelian if and only if $[G,G] \le N$ and this shows that the commutator subgroup of $G$ is the minimal normal subgroup for which $G/N$ is Abelian.
But does this help to prove that "$G$ is Abelian if and only if $[e,e]$ is the only commutator of $G$"?
If yes then how?
Also does there exist any name for $[e,e]$?
Wikipedia claims that the identity element $e$ is the only commutator iff $G$ is Abelian.
$[e,e]=e$ always holds, and yes it has a name: the identity element.
The claim is immediate from the fact that $[a,b]=e\iff ab=ba$, which is clear by the definition of commutator: $[a,b]=a^{-1}b^{-1}ab$.